liv 
if we make for abridgment 
3 mv min’ 
r=\ja(s 2 a r ) ) 
and denote by éay and Es the values which the variation of a, 
and the differential coefficient of that vector taken with respect 
to ¢, are supposed to have at the origin of the time. The defi- 
nite integral denoted here by the letter F is the same which 
was denoted by the letter s in the Essays already referred to, 
and which was called, in one of those Essays, the Principal 
Function of the motion of a system of bodies ; and if we now 
regard it as a function of the time ¢, and of all the final and 
initial vectors, a, a’,-. ay; a’9,-. Of the various bodies of the 
system, and suppose (as we may) that its variation, taken 
with respect to all those veetors, is determined by an equation 
of the form, 
0 = 20F + X(oda — ada, + da. — Say. Gp), (10) 
in which o, o, are vectors, we are conducted, by comparison 
of the coefficients of the arbitrary variations of vectors, in the 
equations (8) and (10), to the two following systems of for- 
mule : 
da , da’ ; 
apts ee 1 ae tae ete (11) 
da, ,da’y _ , . 
mM Ty = % mM! = Oo ss 3 (12) 
of which the former may be regarded as intermediate, and the 
latter as final integrals of the differential equations of motion. 
The determination of the (vector) coefficient a, from the varia- 
tion of the (scalar) function F, is an operation of the same 
kind as the known operation of taking a partial differential co- 
efficient, and may, in these new calculations, be called by the 
same name; but in order to be fully understood, it requires 
some new considerations, of which the account must be post- 
poned to another occasion. 
